On Representations of Semi-Infinite Programs which Have No Duality Gaps

Abstract

Duality gaps which may occur in semi-infinite programs are shown to be interpretable as a phenomenon of an improper representation of the constraint set, u^TP_i ≧ c_i, i ε I. Thus, any semi-infinite system of linear inequalities has a canonically closed equivalent (with interior points) which has no duality gap. With respect to the original system of inequalities, duality gaps may be closed by adjoining additional linear inequalities to the original system. Also, for consistent, but not necessarily canonically closed programs, a partial regularization of original data removes duality gaps that may occur. In contrast, a new “weakly consistent” duality theorem without duality gap may have a value determined by an inequality which is strictly redundant with respect to the constraint set defined by the total inequality system.